Physics & Math Gymnastics

While we enter physics to study the fascinating world of black holes, quarks and the quantum, the brutal truth is that mathematics is the central tool of the physicist. Gauss called mathematics the "Queen of the Sciences", and with good reason. If you don't have a solid grasp of mathematics, you aren't going to get very far.

One thing I noticed when getting my degrees in physics was that many of the students found math to be a painful "aside". In one case that really stands out in my memory, The student in question had thought he was interested in physics but didn't want to bother with the work of physics-which involves diving into the mathematics. But to become a good physicist-or a solid engineer-you need to bite the bullet and become a master of mathematics. It doesn't matter if you're going to be an astronomer, experimentalist, or engineer-in my view if you want to be the best at what you do in these fields, you should have a solid command of math. So if you are interested in physics but aren't a mathematical hot shot, how can you pull yourself to the top of the field? In my view, the answer is to view mathematics the way you would athletics. A friend of mine who shared this view coined the term "mental gymnastics" to characterize his outlook and study habits.

We all aren't Math Genuises

While for some students thinking mathematically comes natural, most of us aren't ready to master the intricacies of studying proofs when we're college freshmen. This article is written for those of us who aren't automatic math whiz kids. If you are a mere mortal who finds math a bit of work, don't be discouraged. It's my belief that average people can raise themselves up to become very good mathematicians with a little bit of hard work. What we need is some training--we need to train our minds to think mathematically. The best way to think about how you can get this done is to draw an analogy between math and athletics.

To master a sport you have to build your muscles and train your body to react in certain ways. For example, if you want to become a great basketball player, you could be lucky enough to be born Michael Jordan. But more likely, you'll have to work at building a basic skill set, and the truth is even players like Michael Jordan put extra work into their craft. Some activiities you might consider that could make you a better basketball player are

  • Lifting weights to build muscle mass
  • Run sprints to improve your ability to run up and down the court without getting tired
  • Spend a large amount of time shooting free throws, doing layups and practicing basic skills like passing

It turns out that becoming a successful physicist or engineer is in many ways similar to athletics. OK, so suppose you want to study Hawking radiation and string theory, but you are not a hot shot mathematician and weren't the best student. Instead of just reading a bunch of books or lamenting the fact we aren't an Einsteinian genius, what are the mathematical equivalents to lifting weights or running sprints we can do to improve our mathematical ability? In my view, we can begin by following two steps

  • learn the basic rules first
  • repeat, repeat, repeat
That is do tons of problems. In my view a student should start off simple. Don't try to understand the proofs. For example, in my recent book, "Calculus In Focus", I take the perspective that students need to learn math by following the formula: show, repeat, try it yourself. That is

  • Show the student a given rule, like the product rule for derivatives
  • Focus on mastering calculational skills first. Do this by showing the student how to apply the rule with multiple examples.
  • Repeat, repeat, repeat. Do a given type of problem multiple times so that it becomes second nature.
Once the "how" to solve problems is second nature, then go back for a deeper look at the material. Then learn the "why" and start learning the formality of mathematics through proofs and theorems. I use this approach to drill the central ideas of calculus in my book Calculus in Focus. More information can be found 

In addition to the basic approach, a certain baseline has to be established if you want to build yourself up for a formal career in math, physics, or engineering. Let's build up a fundamental skill set that is going to build your fundamental math skills and help you master any subject. A few key areas I think students should focus on are outlined below.

The Importance of Algebra

If you study physics or engineering, algebra never goes away. So the first step on the road to becoming the next Stephen Hawking is to master this tedious yet fundamental subject. Do yourself a favor and pick up a decent algebra book and work through it. Do every problem so that by the end of the book, factoring equations, logarithms and other math basics are second nature for you. In the same way that lifting weights is going to make a football or basketball a better athlete when the games are actually played, mastering algebra will pay off later when you're doing your homework in dynamics or quantum theory.


If you go on to become an electrical engineer and study circuit analysis or decide to master black hole physics, one fundamental area of business you'll have in common with your colleagues is trigonometry. Make sure you know your trig inside and out, learn what the trig functions really mean and master those pesky identities. Also don't over look this one crucial fact-trigonometry also provides a simple arena where you can learn how to prove and/or derive results. We all know that later, when you take advanced physics courses, you're going to see the words "show that" pop up frequently in your homework problems.
 This is sure to cause headaches among the mere mortals amongst us, but it turns out you can improve your skills in this area in a non-threatening way by deriving trig identities. Instead of viewing the derivation of trig identities as a tedious obstacle, start to look at this as an opportunity. All trig books have homework problems where you have to derive an identity so pick up a trig book and do it until your blue in the face. Take it seriously and write up each proof as if you were submitting a short paper to a major journal. This will teach you how to go from point A to point B mathematically and how to write up a derivation in a formal way that will allow someone else to understand what's going on. If you do, later it will be easier to get through homework in advanced classes, you'll get better grades, and you'll develop a good foundation for writing up theoretical derivations for research papers.

Graphing Functions

While any function can be graphed easily on the computer or on a graphing calculator, it is very important to be able to graph a function on the fly with nothing more than a pencil and paper. The key abilities you want to focus on are developing an intuitive sense for how functions behave and learning how to focus on how functions behave in various limits. That is, how does a function look when the argument is small? How does it behave as the argument goes to infinity? Dig out your calculus book and review techniques that use the first and second derivative to graph a function. I review these extensively in my recent book "Calculus in Focus".

Series and Complex Numbers

In my opinion, understanding the series expansion of functions and the behavior of complex numbers can't be underestimated. If you want to understand physics, you need to master the use of series. Start by learning how to expand a function in a series. Some series should be second nature ('oh yeah, that's cosine"). Learn about convergence. Get a copy of Arfken and review the solution of differential equations using series. Try to get an intuitive feel for cutting a series off at a given term while retaining the essential behavior of the function. These are tools that are important when studying theoretical physics or advanced engineering. 

Learning physics should be easy

While its true that not all of us are Einsteins, should it be so difficult to learn math, science, and engineering that only a small handful of people can get degrees in these fields?Part of the problem is the way that math, physics, and engineering are taught. I haven't decided if there is a conscious conspiracy or not-but the truth is these subjects are generally taught in a way that is not helpful to most people. Maybe its because the vast majority of people that become professors are simply quite a bit smarter than the rest of us, and they don't realize what they're doing because they just "get it" and figure if you don't "get it" you are'nt cut out to be a physicist or mathematician.

In a typical college experience, I took "Electrodynamics" in graduate school. The professor was a great lecturer, but his lectures were really a complete waste of time. Basically, we spent our days in class listening to him spit out the book. He would recite the theorems and prove them. Had the book not been available his lectures would have been gold, but since we could buy a book, in fact since we were required to buy a book that had all this exact material in it, the lectures turned out to be no help at all. It's important to reinforce concepts to be sure, but physics, math and engineering are about doing things. These are active fields where problems must be solved. Its not about memorizing a theorem, its about applying it or being able to derive a new one.

Rather than "lecture", I would prefer that professors assign a book they are going to follow and then use the class time to help students solve problems. They should have the students read a given chapter before coming to class, and then spend class showing students how to solve some problems. Homework can then be assigned allowing students to build on what they did in class to learn how to solve the problems on their own.

Maybe physics professors like having a realm of mystery surround them. They like to feel smarter than everyone else and often aren't interested in helping people learn. So they keep problem solving tricks to themselves, and then tell the students who don't "get it" that they should become experimentalists or engineers. In Quantum Mechanics Demystified I have attempted to provide readers with format that makes learning physics straightforward. I show you how to solve quantum mechanics problems, and then you can try to do similar problems on your own.

From the fringes of quantum physics and relativity theory comes Bob Lazaar, an interesting man who claims to have worked on UFO's at Area 51. Tonight on coast to coast AM, the show will be guest hosted by George Knapp, a TV reporter who broke the Bob Lazaar story several years ago.

Lazaar is a very well spoken and charismatic man, but his story doesn't add up. I am not even talking about his wild claim to have worked on UFO's and "antigravity" propulsion at Area 51. Let's just start with his basic storyline.

He claims to be a physicist that worked at several places where a clearance is required, including Los Alamos and Area 51. I don't recall where he claimed to have gone to school, it might have been Cal Tech or MIT, but something that stood out for me was he claimed not to recall any professors names.

Anyone who has gotten a technical degree will recognize this claim as absurd. Students in math, physics and engineering run into hard professors and nutty professors, and good professors that just downright torture you during the semester. At least some of the names of these professors stick to you like glue throughout your life and its something that binds you to your fellow students at the institution where you got your degree. So when I heard Lazaar make this statement it struck me as odd to say the least.

Then there is the problem of his academic record. As I recall he claimed to have an advanced degree in physics yet there was no record of him having attended any of these prestigious instituions. This was explained by the claim that the CIA wiped out his record or something like that.

It has also been difficult to verify his work record. The sole evidence he worked at Los Alamos is a single paystub for a 2 week period where he worked as a technician. Again, I believe this is explained away by the vast powers in the CIA wiping out his record.

Anyone who believes the CIA or any government entity is that powerful or that places like Los Alamos are that secretive has been spending too much time watching television! The fact is its no secret who works at Los Alamos or any other government lab. The only things that are secret are the details of the project they work on. Its easy to find out that Joe Schmo works at X national lab in department Y. To me, the fact that no such record exists for Lazaar indicates that at best he worked as a temp here and there doing contract work. He was not some top scientist that would be called upon in the extremely unlikely event that they needed someone to "reverse engineer" a UFO. In a nutshell, I basically don't buy into Lazaar or his crazy story.

If anything, the Lazaar phenomenon is an interesting study in human behavior and our wish to live in a universe inhabited by aliens. These are all scientific issues and having a reasonable understanding of them is important. The population should have their own understanding of the issues to a certain degree rather than having to rely on experts for everything. This can only be done through the education system, and while these are conceptually based topics ultimately there is a mathematical underpinning. I am not suggesting that people should be going out and doing their own calculations of say uranium half-lives, but doing some calculations like that in school will allow someone to make more reasoned judgements on many issues-like storing nuclear wastes at Yucca Mountain.

Cohen says that algebra isn't a high or the highest form of human reasoning, and that writing is. Frankly Mr. Cohen I beg to differ. Mathematics is the highest form of human reasoning and is the basic underpinning of our modern society. It transcends the sciences, being at the root of the human genome project, the design of lasers, electric power, radio and cell phones and the internet. In short the entire modern world is fundamentally mathematically based. Writing was already highly developed long before calculus came into existence.

Cohen also claims that a computer or calculator can do math while they can't write. Cohen's understanding of how computers are used in mathematics is naive. In higher mathematics, doing a solution like the quadratic equation is trivial. Its the understanding of the solutions and properties of equations that require higher reasoning. The computer is used to churn out solutions to equations that are too hard for the human mind to solve directly (no analytic or closed form solution). In the end a human being has to analyze and interpret the results-something a computer can't do.

One of the themes touched on in "Equations of Eternity" by David Darling is the unreasonable effectiveness of mathematics in describing the physical world. Time and again, as Darling points out, mathematicians have worked on some obscure theoretical idea or area that seems to have nothing to do with reality. Then years later physicists stumble on it and discover that it describes some physical process in absolute detail, down to the last dotted i and crossed it.

A great example of the connection between mathematics and the physical world is the discovery by Maxwell of well, Maxwell's equations. During the 19th century the frontiers of science were being pushed by people studying electromagnetic phenomena. Years earlier Coloumb had figured out how to describe the electric force between two charges. In the early to mid-1800's physics had moved quite a bit beyond that to consider electric currents and magnetic fields. It was here that mathematical insight would prove to be an unusually effective tool-revealing properties of nature hidden to the senses.

In about the 1830's Ampere worked out a "law" that relates the magnetic field to a flow of current. Ampere's law has a very precise mathematical form which was worked out from careful experimental observation.

These "laws" of vector calculus are abstract mathematical laws--supposedly laws of pure thought. At first sight one might not expect that they would hold precedence over experimental observation. But it turns out they do. Maxwell used the laws to determine what form Ampere's law should really have, and in the process discovered something that was unknown at the time-radio waves.

This is just one small example of the interplay between math and physics. Later we'll explore connections between abstract mathematics and quantum theory which describes every last detail of atomic behavior.

The Holy Grail of physics is the unification of quantum physics and relativity, a Herculean task trying to wed together two spheres as different as night and day. On one hand we have the world of the very large. This is the world of stars, planets and galaxies-the world governed by Einstein’s relativity. On the other hand we have the world of the very small-the world of atoms, neutrons, and quarks-governed by the quantum. Each of these two realms not only describes different types of objects or different sized objects-they require different types of mathematics. Even worse-the world of stars and galaxies seems to be governed by a classical, deterministic physics which fits neatly into a beautiful geometric theory, while the world of elementary particles is governed by probability, randomness, and mysterious mathematical worlds called Hilbert spaces-the world of the quantum dice.

At first glance these two theories can hardly be thought to be describing objects that belong to one and the same universe-but they do exactly that. Stars are made of atoms that obey the laws of quantum physics. Out of chance, chaos and ghostly entanglement-the orderly structure of a galaxy somehow emerges-and if we look at the ghostly quantum particles closely-atoms and elementary particles do fall in a gravitational field. Therefore there must be a path forward to a unified new physics.

Before embarking on the path of unification, it is important to make sure that one has a complete and thorough understanding of those two pillars of physics that are already well established-quantum physics and general relativity. This understanding is necessary before moving on to explore efforts at unification such as string theory and loop quantum gravity.

Let’s take a stab at quantum theory. In my book Quantum Mechanics Demystified, I lay out the mathematical framework of quantum theory. But what is the conceptual framework-the basic building blocks that one wants to come away with before trying to put together a unified theory that describes the universe?. 

These are the ten keys to quantum physics. They may not be the only ones-I am simply making suggestions of key concepts. You may wish to add your own. Before we start-a brief note on notation. We will denote the state of a particle or system with a bold capital letter, such as F or G, while a scalar (a plain old number) will be denoted by an small italic letter such as a or b.


One principle that plays a central, absolutely vital role in quantum theory is the notion of superposition. Imagine if you will that a particle can be in two mutually exclusive states that we denote F and G. These two states could represent going through one slit or the other in the two slit experiment, or they could be two different energy states of an electron in an atom, for example. The principle of superposition tells us that the state formed by their linear combination-i.e. their sum-is also a valid quantum state. That is a system can be in the state described by H = F + G. A quantum superposition is a special sort of beast-when we look at the system, that is when we make a measurement, we never find it in some strange mixture of the states F and G. Rather, it is always in one state or the other. That is when the system is prepared in state measurement will sometimes find the system in state F, while at other times, measurement will find the system in state G. Note that the numbers a and b can be complex. In key #2, we interpret the meaning of the state and explore it further.


The Born Rule tells us the probability of finding a quantum system in this state or that using a simple recipe. If a quantum system is in the state described by a F + b G, then the probability that the system is found in state F when a measurement is made is found by squaring a while the probability that the system is found in state G when a measurement is made is found by squaring b. It is important not to confuse the fact that the Born rule tells us how to extract probabilities from a quantum state with the notion that the state is a mere statistical mixture. A superposition state like leads to interference effects-like the fringes seen on the screen in the double slit experiment-something a statistical mixture can’t do.


A key concept in classical science is that if you set up an experiment in exactly the same way several times- you will get repeatable results. The probabilistic nature of quantum theory-which is inherently fundamental and is not due to a lack of precision in our measurement devices-means that in many cases, if we prepare several systems in a given state, we will get different measurement results when the experiment is run. Once again, suppose that we prepare the system in the state a F + b G. If we run the experiment 4 times, we might make measurements and find the results FFGF. The next day if we again prepare systems in state a F + b G , we might instead get FGGG. Now if we do the experiment a large number of times, then the relative fractions of F and will tend to the probabilities given by squaring the coefficients a and b.


The Heisenberg uncertainty principle tells us that we cannot know the values of two complimentary observables with absolute precision. The quintessential example given in most textbooks is the uncertainty relation between position and momentum. In short, the uncertainty principle tells us that the more precision we use in measurement of position, the less we can know about momentum and vice versa. If we wish, we can know the value of one variable to any precision we like-but at the expense of complete uncertainty in the other variable. For example, if we choose to measure a particle’s momentum with great accuracy, then we sacrifice knowledge of the particles position. The uncertainty principle signs the death warrant of classical, deterministic physics.


Quantum states to evolve in time in a deterministic manner that is governed by the Schrodinger equation. Or in a more modern sense-the time evolution of quantum states is described by unitary evolution.


The fifth key to quantum theory is that the world of quantum mechanics lies in the mathematical realm of vector spaces. This means that the mathematics of quantum physics is the mathematics of linear algebra. If you want to master quantum theory then you need to know linear algebra. Learn how to manipulate matrices, how to calculate determinants, and how to find eigenvectors and eigenvalues. Learn about abstract vector spaces. Finally, learn about special types of matrices, in particular Hermitian and Unitary matrices.


This follows on the heels of point 6. In quantum theory, physical observables like momentum and energy are represented by operators. When considering the wave function approach of the Schrodinger equation, operators are instructions to do something to a function-compute the derivative say. In the mathematically equivalent matrix mechanics derived by Heisenberg, operators are represented by matrices.


Following key #7, since physical observables are represented by mathematical operators, the next logical question to ask is what are the possible measurement results as predicted by the theory? It turns out that these are the eigenvalues of the operators used to represent physical observables.


An important issue in quantum theory is the following. Is the wavefunction a real, physical entity? Or does it just represent our state of knowledge? The meaning of the wavefunction is an important issue to resolve before we can have a “theory of everything”.


Quantum systems interact with their environment. In doing so, the strange quantumness of their nature is lost. That is superposition and the interference that comes along with it is washed out by interactions with the environment. It is believed by many that this is how the classical world of our senses arises out of the quantum morass.


Quantum physics is filled with mysteries and perhaps the greatest mystery of all is that of entanglement-the spooky action at a distance type correlation originally put forward by Einstein and his colleagues back in 1935. If two particles A and B are entangled, then their properties become correlated. In a certain state, if you measure the spin of A and find it to be spin-up, then the spin of B is spin-down. Or if you measure A and find it to be spin-down, then B is spin-up. If that is too abstract to wrap your mind around, for a loose analogy imagine that A and B are entangled dice correlated such that they always roll the same number. Roll A and find a 3, then rolling B is guaranteed to give a 3.

Spooky action at a distance tells us that we can leave A here on earth and carry B all the way to the other side of the galaxy-and their measurement results will remain correlated. Of course this isn’t completely worked out, even assuming the particles could be protected from the environment, relativistic effects and gravity might hamper such a scenario, but spatial separation, i.e. distance alone doesn’t seem to have an effect on entanglement.

The type of correlation that results in entanglement is a bit spooky in itself. It’s not really mere correlation. Without diving into the mathematics, suffice it to say that basically the two particles loose the essence of their individual identities. In an entangled system, the system as a whole assumes an identity. 

It is as if the whole becomes greater than the sum of its parts. To understand the entangled system you need to understand the whole and cannot understand it from the individual components alone. This amazing phenomenon-which has been confirmed in the laboratory-is of central importance in quantum computation and quantum cryptography. Quantum physics can be mathematically daunting. Before you go on to tackle string theory and quantum gravity-make sure you have these central ideas down

Evidence to House of Commons Sci Tech Select Committee on Research Integrity

Sorry not to be in regular blogging mode at the moment. Here’s a video of our evidence session to parliament, where they are running an inquiry into research integrity. I think clinical trials are the best possible way to approach this issue. Lots of things in “research integrity” are hard to capture in hard logical rules,  so you end up with waffly “concordats” and rules that are applied inconsistently. With clinical trials you can make clear rules, you can measure compliance, and you can enforce compliance. There is lots of chat about this in the video below from 17:37 with me, Simon Kolstoe (who did this neat audit using ethics committee data), and Sile Lane 
It’s a very interesting inquiry overall. In the session before you can see Ivan Oransky from RetractionWatch and others. You can read more about the inquiry overall here, I’ve posted my submission below. Cheers!
Written evidence submitted to House of Commons Science and Technology Committee inquiry on Research Integrity, March 2017
Dr Ben Goldacre
I am a medical doctor and academic. I run the Evidence-Based Medicine DataLab at the University of Oxford, with a small team of ten people, where we make tools that turn medical data into actionable insights. I also work on public engagement, wrote the “Bad Science” column in the Guardian for a decade, and have sold over half a million books on problems in science and medicine. I am a co-founder of, the global campaign for all clinical trial results to be reported. In policy work I conducted an independent external review on evidence-based policy for DfE, co-authored a Cabinet Office paper on randomised trials in policymaking, and have sat on various advisory groups for MoJ, NHS England, and DfE.
 Executive Summary
Fraud is not the most important issue. The culture of incomplete and inaccurate reporting of research has greater impact on patients and society, and can be addressed. Clinical trials are the “canary in the cage”. Trials are large expensive research projects used to generate knowledge that is then used, in clinical practice, to make vitally important decisions; and yet trials are commonly left unreported, or misreported. This is a waste of money, and distorts the evidence underpinning medical practice. Non-reporting of clinical trials has been left unaddressed for three decades. The government could take simple action: require all trials to be reported; support audit that identifies which trials have been left unreported; and fund research into what will improve standards. Improvements and innovation in this area will generalise to the rest of science.
 The extent of the research integrity problem
1.1 Fraud is not the most important issue.
Fraud is problematic, and our mechanisms to identify and address it remain poor; however fraud is also relatively infrequent, and universally recognised as unacceptable. More urgent is the widespread culture of incomplete and inaccurate reporting of scientific research; the culture of permissiveness and denialism around this; and the failure of government, public funders, and public institutions to take simple steps to improve the situation.
 1.2 Non-reported studies.
The results of clinical trials are used by clinicians, patients, health services and policymakers to make informed choices about which interventions are most effective. However there is extensive and longstanding evidence that the methods and results of clinical trials are routinely left unavailable. To assess non-publication researchers take a list of trials known to have been completed; they then check some time later whether the results have been made available, usually by searching academic journals (where doctors and researchers would generally expect to find trial results), or sometimes by searching in other less well known repositories. A 2014 systematic review1 of all the literature on this question summarises 17 studies following up cohorts of trials approved by ethics committees: overall 46% of trials were published. They also found 22 studies on cohorts of trial listed in public registries: 54% were published. Studies with statistically significant results were three times more likely to be published, which is consistent with previous work2.
 Furthermore, it is impossible to get a perfect assessment of how many trials are unreported, because we do not know what trials have been conducted: a 2015 cohort study on all drugs approved in 2012 found that only 57% of trials were registered3; and in our team’s experience of conducting similar research, ethics committees can refuse to disclose their list of trials that have been approved. Furthermore we have repeatedly found evidence that the information entered into trial registries is incomplete and inaccurate.
 1.3 Mis-reported studies
Related to the problem of non-reported trials is that of mis-reported trials, and the selective reporting of results from within one study. It is very common for researchers to record a large number of different outcomes during the conduct of a trial (such as blood tests, hospital admissions, symptom rating scales, and so on); and then only report some of the results at the end. This is an important source of bias, and can lead to exaggeration or misrepresentation of the true effects of a treatment.  
 There are various systems in place to prevent this practice. I will describe these systems and their failure in some detail, as they represent a clear example of how we have failed to address shortcomings in research, despite extensive discussion, guidelines, and investment in infrastructure. Before carrying out a clinical trial, all measured outcomes are supposed to be pre-specified in a publicly accessible trial protocol, and the trial’s registry entry. The trial report is supposed to contain results for all these pre-specified outcomes, or declare and explain any deviations. The importance of reporting all prespecified outcomes and documenting changes is emphasised in numerous authoritative locations including the International Conference on Harmonisation of Good Clinical Practice (ICH-GCP) 4, and the CONSORT guidelines on trial reporting5 which are widely respected and endorsed by over 500 individually named academic journals6. It is one of the core purposes of trial registration, itself supported by organisations including the WHO, the International Committee of Medical Journal Editors, and various national laws and regulations.
 However, despite near universal recognition of the importance of complete outcome reporting, trial reports in academic journals routinely breach these norms. A 2015 systematic review 7 found 27 studies comparing prespecified outcomes against those reported, in cohorts of between 1 and 198 published trials. The median proportion of trials with a discrepancy on primary outcomes alone was 31%.
 Because over 500 journals have committed to ensure all trials are correctly reported, and yet the same journals continue to mis-report trials, we ran a project monitoring every trial in five major journals, and sent a correction letter, for publication, on every mis-reported trial. This was not to repeat the finding that trials are misreported, but rather to assess whether the self-correction mechanisms of science are operating correctly. We found that they are not. Most letters were rejected, and the responses from trialists and editors demonstrated that many do not understand the issue of correct outcome reporting, or are not concerned by breaches, despite their public stance to the contrary, and despite all journals in our study being listed as endorsing the CONSORT guidelines, which require complete outcome reporting. I am happy to share more details on this project prior to its final publication, all data is shared online 
 The impact of these ongoing problems
We have collectively failed to address these issues throughout the ecosystem of science and medicine: this includes funders, regulators, researchers, journal editors, professional bodies, and more. The immediate burden is clear. In pure science, research spend is wasted. In applied sciences such as medicine, the impact is more acute. We cannot make informed choices about which treatment works best when the results of clinical trials are routinely and legally withheld from doctors, researchers, and patients. There is also an equally important reputation hit to science. The public are increasingly aware that serious problems have been left unaddressed: that trial results are routinely withheld, that there has been little serious effective effort to fix the issue over decades, that the biggest players in the ecosystem of scientific research are not taking adequate action. Faced with this reality patients and the public will conclude, to a greater or lesser extent, that science is structurally incompetent, or even somewhat corrupt. Our permissiveness and failure to put our own house in order lays fertile ground for quacks, anti-vaccination conspiracy theorists, and climate change denialists. Science needs to earn and retain its reputation for generating accurate knowledge about the world.
 The effectiveness of controls/regulation (formal and informal), and what further measures if any are needed
 Neither soft “culture change” interventions nor legislation have been effective to date. Sharing the methods and results of all trials has been recognised as an ethical and scientific imperative for many years8–10. More recently institutions such as the World Health Organization (WHO)11, the European Commission12, and the US Food and Drug Administration (FDA)13 have called for results disclosure. The first data on non-publication in medical trials was published in 1986 9; and yet three decades later the data shows that trial results are still routinely left unreported. In the US, the FDA Amendment Act 2007 (FDAAA) requires sponsors to post results within 12 months for a subset of clinical trials. Two cohort studies have now explored compliance with the broad intent of FDAAA and found compliance rates of only one trial in five 14,15; however, the formal implementation of this legislation has been delayed by a decade, and the first trials to be legally covered by it will not complete until 2018. The UK government could address this very simply, with the steps set out below. There are innumerable regulations, edicts, reports, guidelines, and strategy documents around trial reporting. None have been enforced or implemented, and breaches are not documented. This creates a dual burden: it creates the illusion that problems have been fixed, when in reality they are ongoing, and so removes the sense of urgency around improving standards, leaving a false sense of reassurance.
 What can be done.
We have seen poor progress in addressing these issues. In my view this is driven by a small number of key factors, all of which can be reversed.
 4.1 Generate better data on breaches
We can only improve quality by measuring performance. Trial publication is a very simple thing to measure. We conducted an audit of publication of all trials in Oxford and helped identify areas for improvement16.  Our team is also creating a number of global “Trials Tracker” dashboards comparing publication performance by various metrics (although this is currently made needlessly difficult by poor quality information on trials in databases and registries). Audit is a positive process. It identifies systemic problems, and regions where more targeted education or assistance is required. It identifies areas where systems are functioning well, so that others can learn from them. The very process of public audit itself can also help to drive up standards: there is currently almost no accountability for those who do not report their trials, or mis-report their trials, whether they are publicly funded, or funded by industry.
 The government could and should simply require all UK research funders to publicly declare all trials they have funded, and whether those trials have published their results, with performance statistics reported, and all individual reported and unreported trials identified. Several funders committed to produce summary statistics on the proportion of trials reported, among those they have funded, at this Committee’s previous inquiry into non-publication of trial results. To my knowledge, four years later, none of these audits have been made publicly available.
 The government could also require all ethics committees to generate data on non-reporting with no great increase in administrative burden. At present, all trials in the UK need to be approved by an ethics committee. Currently ethics committees approve research, and then effectively lose contact. The government could simply require all ethics committees to request an end of study report on all approved trials, and publicly disclose whether each study is reported. In related work, we have found that ethics committees are reluctant to share their deliberations on trials with questionable designs: this material should also be routinely publicly available17. Overall, we should have a national programme of audit on all clinical trials conducted in the UK, and report performance on whether trials are reported, broken down by institution, funder, researcher, disease area, and so on18. This could be conducted by institutions, or ideally independently.
 It is worth noting that many US universities now have staff nominated to work on ensuring that all trials are registered, with their results reported within 12 months of completion. These staff take responsibility for monitoring compliance, and supporting researchers who require assistance. They are organised under an umbrella organisation: the “Clinical Trials Registration and Results Reporting Taskforce.” The government should require UK universities and funders to replicate this model.
 4.2 Create a clear legislative requirement to register and report all trials
The UK government should require that all trials conducted in the UK are registered, and their full methods and results publicly reported. There is new EU legislation requiring this, but only for drug trials, and the UK is leaving the EU. Previously it has been suggested that taking a strong line on this issue would undermine our ability to attract international clinical trials business to the UK. This is mistaken. The UK share of the global clinical trials market is shrinking annually because of cheaper options in low- and middle-income countries, and Eastern Europe. We cannot compete on price. We should not even consider competing on weaker regulation. We can, however, compete on quality. Good quality evidence is about rigour, and the credibility that comes from this rigour. If we make it clear that UK trials are the most reliable in the world – that they produce more reliable evidence than other territories – then we will attract trials, and help drive up standards for science globally.
 4.3 Legitimise and Fund Work on Research Integrity
Working on issues around research integrity is currently regarded, in derisory terms, as either a “hobby” outside a researcher’s real work; or as somehow radical, transgressive, or trouble-making. This dismissive attitude reflects negatively on the profession. The situation arises, in part, because there is essentially no funding in the UK to work on problems around research integrity. That is a startling omission for an issue so central to the whole of science: it reflects badly on the discipline and the community, and is surely connected to the very poor progress seen across all these issues. Without funding on research and interventions to improve research integrity, we cannot be surprised that the problems persist.
 The Committee could improve this situation by communicating clearly that work on research integrity is a legitimate academic pursuit; and by encouraging government agencies and public institutions to cooperate with efforts to document the extent of the problem. If we are to make concrete progress, the Committee should also request funders to to create specific funding streams on research integrity. This would help to achieve five important goals:
 Provide funding to develop and assess new interventions to improve research reporting and research integrity
  • Provide pilot funding to develop infrastructure to improve reporting, such as “Registered Reports”, or new reporting platforms.
  • Provide implementation funding for interventions shown to improve reporting such as the guidelines of EQUATOR network, and related pro-active offshoots.
  • Support research to better document the extent and character of the problem
  • Support independent audit to compare individual institutions and sponsors, and drive up standards by identifying areas requiring the most improvement
Research integrity is a global problem, and work in this area has positive global impact. If we do not fund work on research integrity we must expect all the problems documented by the Committee to persist. This is an active choice by government and funders.


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